MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  imass1 Structured version   Visualization version   GIF version

Theorem imass1 6105
Description: Subset theorem for image. (Contributed by NM, 16-Mar-2004.)
Assertion
Ref Expression
imass1 (𝐴𝐵 → (𝐴𝐶) ⊆ (𝐵𝐶))

Proof of Theorem imass1
StepHypRef Expression
1 ssres 6012 . . 3 (𝐴𝐵 → (𝐴𝐶) ⊆ (𝐵𝐶))
2 rnss 5941 . . 3 ((𝐴𝐶) ⊆ (𝐵𝐶) → ran (𝐴𝐶) ⊆ ran (𝐵𝐶))
31, 2syl 17 . 2 (𝐴𝐵 → ran (𝐴𝐶) ⊆ ran (𝐵𝐶))
4 df-ima 5691 . 2 (𝐴𝐶) = ran (𝐴𝐶)
5 df-ima 5691 . 2 (𝐵𝐶) = ran (𝐵𝐶)
63, 4, 53sstr4g 4025 1 (𝐴𝐵 → (𝐴𝐶) ⊆ (𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wss 3947  ran crn 5679  cres 5680  cima 5681
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5149  df-opab 5211  df-cnv 5686  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691
This theorem is referenced by:  predrelss  6343  vdwnnlem1  16963  dprdres  19984  imasnopn  23593  imasncld  23594  imasncls  23595  utoptop  24138  restutop  24141  ustuqtop3  24147  utopreg  24156  metustbl  24474  imadifxp  32390  esum2d  33712  eulerpartlemmf  33995  bj-imdirco  36669  brtrclfv2  43157  frege97d  43182  frege109d  43187  frege131d  43194  hess  43210  resimass  44615  setrecsss  48132
  Copyright terms: Public domain W3C validator